Harvest Moon: A Wonderful Life Special Edition

Healing a sick animal involves buying animal medicine from the ledger for 200g per medicine. The trophy will unlock on the next day, not when you administer the medicine. I don't recommend letting your cow get sick during Ch. 1 because you can't milk her while she's sick, thus losing money for it. You could do a seperate save for this trophy, or simply wait until later on in the game.

If you decided to wait until later on, there's a good chance that one of your animals will get sick once you start just sleeping through the remaining years of the game because you finished up all the other trophies. Keep an eye on them until one of them is sick and give them the medicine. It doesn't matter if it's a chicken, sheep, goat or cow.

For me, my goat got sick twice towards the end of Ch. 2, despite receiving the same treatement as my cows (they spent most of their time outside unless it was raining/snowing), so that might happen for you as well. If not, most of my animals died once I started power-sleeping, so I could have healed any one of them before their death to unlock the trophy.

If after reading all that, you still want to get this trophy as soon as possible, simply leave your animals outside at all times, especially during rain and don't feed them. Make sure the grass in your pasture is always cut so they can't eat it. Eventually, one of them will get sick (most likely the god damn goat).